Impact of Thermodynamic Non-idealities and
Mass Transfer on Multi-phase Hydrodynamics
M. Irani
1;2, R. Bozorgmehry Boozarjomehry
1and M.R. Pishvaie
1;Abstract. Considering the non-ideal behavior of uids and their eects on hydrodynamic and mass transfer in multiphase ow is very essential. Simulations were performed that take into account the eects of mass transfer and mixture non-ideality on the hydrodynamics reported by Bozorgmehry et al. In this paper, by assuming the density of phases to be constant and using Raoult's law instead of EOS and the fugacity coecient denition, respectively, for both liquid and gas phases, the importance of non-ideality eects on mass transfer and hydrodynamic behavior was studied. The results for a system of octane/propane (T = 323 K and P = 445 kPa) also indicated that the assumption of constant density in simulation had a major role to diverse from experimental data. Furthermore, comparison between obtained results and previous reports indicated signicant dierences between experimental data and simulation results with more ideal assumptions.
Keywords: Multiphase ow; VOF; Mass transfer; Raoult's law; Non-ideal thermodynamics; CFD.
INTRODUCTION
Many processes in chemical and petrochemical indus-tries involve gas-liquid mass transfer in the gas or the liquid phase. Despite this fact and substantial research eorts devoted to understanding detailed knowledge on uid ow, mass and heat transfer and thermodynamic behavior, as well as their interactions, are still lacking. In the past decade, the engineering community has been active in exploring the possibilities of utilizing Computational Fluid Dynamics (CFD) in the modeling of multiphase ow phenomena. However, in most of the studies, mixture non-ideality and mass transfer are not considered simultaneously.
The direct application of CFD to chemical pro-cesses faces several problems, however, even in single phase ow, ow and mass transfer are described by highly non-linear terms that often cause numerical instabilities. More complex phenomena, such as
multi-1. Department of Chemical and Petroleum Engineering, Sharif University of Technology, Tehran, P.O. Box 11155-9465, Iran.
2. Research Institute of Petroleum Industry, Tehran, P.O. Box 18745-4163, Iran.
*. Corresponding author. E-mail: pishvaie@sharif.edu
Received 1 July 2009; received in revised form 17 November 2009; accepted 6 March 2010
phase ow and interfacial mass transfer with rigorous non-ideal behavior, are encountered in multiphase ow in chemical processes [1].
Recently, \hybrid" approaches have emerged as an alternative. In those, CFD is employed only for hydrodynamic simulation, while the chemical phenom-ena are resolved in a custom-built compartmental model [2]. Bauer and Eigenberger [3,4] used a \zone model" to study a bubble column reactor; Bezzo, Macchietto and Pantelides [5] developed an interface of communication between the gPROMS modeling software and a commercial CFD code. Zauer and Jones [6] used a segregated feed model, in conjunction with CFD, to study precipitation in a stirred tank.
A fundamental weakness of all multi-zonal models is the diculty of characterizing the mass and energy uxes between adjacent zones. This is the case because the uid properties are functions of system conditions (e.g. composition, temperature and pressure). How-ever, this framework is only applicable to systems of which their physical properties are relatively weak functions of intensive properties [6].
Krishna and van Baten [7] have studied the inter-phase mass transfer and reaction (rst order reaction rate) for one species without considering mixture non-idealities. In their study, densities were constant and they had estimated equilibrium constants with Henry's
coecients. They had also neglected the eect of mass transfer on the hydrodynamics behavior of the system.
Later, Breach [8] modeled non-ideal vapor-liquid phase equilibrium and mass and energy transfer in a binary system (H2O, H2O2). Because of operating
conditions in his work, (P = 100 kPa and T = 433 K) he neglected non-idealities in calculations of liquid density and gas phase equilibrium calculations. He also ignored the eect of non-idealities on the calculation of gas and liquid phase internal energies.
Also, Banerjee modeled the evaporation of a binary mixture of ethanol and iso-octane into air owing in an inclined 2D channel. Simulation has been carried out at atmospheric pressure, and temperatures ranging from 300 to 340 K. The liquid phase density was calculated based on the averaged mass fraction of individual components, and the gas phase has been considered as ideal gas. He considered two-phase cells as an interface in which the gas and the liquid are in equilibrium. Therefore, the size of meshes should have been very ne around the interface [9].
Recently, the eects of non-idealities on hydro-dynamic behavior have been studied based on a CFD framework [10]. The properties of each phase have been rigorously modeled as a function of tempera-ture, pressure and concentrations. The VOF interface tracking method has been used for multiphase ow considerations. Mass transfer during condensation and vaporization was modeled by chemical potential at the liquid-vapor interface. Mass transfer resulting from the chemical potential eld is determined by T P ash calculation at the liquid-vapor interface. The equilibrium calculations were performed using the fugacity coecient denition for both the liquid and gas phases through an equation of state.
In the present article, two comparative cases (Cases 1 and 2) were considered. The obtained results were compared with a case (Case 0) presented by Irani et al [10]. In the rst case (Case 1), the densities of phases were assumed to be constant, and in the second case (Case 2), they were calculated by the Peng-Robinson equation of state. In both cases, Rault's law was used instead of a fugacity coecient denition for both the liquid and gas phases, in order to study the aection intensity of non-ideality on hydrodynamic behavior (Table 1).
The mathematical model of the system is
de-scribed in the next section (Mathematical Model), and the benchmark used in this study is explained afterwards in the section of \Benchmark for the Val-idation of Simulation". The last section goes through the comparison between simulations and experimental results.
MATHEMATICAL MODEL
Continuity Equation for the Liquid and Gas Phases
The variation of liquid holdup with time and position is obtained by solving the continuity equations for the liquid and gas phases. The continuity equation for the owing liquid and gas is written in terms of accumulation and convection terms, balanced by the total mass transferred to and from the other phases.
Since gas and liquid phases do not interpenetrate each other in the reactor, the VOF approach is used. In this approach, the motion of all phases is modeled by formulating local, instantaneous conservation equa-tions for mass and momentum transfer [11].
The continuity equation for phase, `q', in a mul-tiphase ow problem is as follows:
@
@t(qq) + r:(q~vq) = Spq;
Sqp= Spq: (1)
The velocity vector, ~, comes from solving the Navier-Stokes Equations (NSE). The right-hand (Spq) side
describes mass transfer from phase p to q where q is
the volume fraction of phase q, which needs to satisfy the following relation:
n
X
q=1
q = 1: (2)
One of the most important characteristics of a multi-phase system is the fractions of various multi-phases. Thus, it is necessary to know the volume fraction, q of each
phase, q, in the entire computational domain. Momentum Transfer Equations
The variation of velocity with time and position is calculated by solving the momentum balance equation.
Table 1. Comparison between case studies considered in this work.
Case Studies Phase Density Phase Equilibrium Calculation Method Comments Case 0 EOS Fugacity coecient for gas & liquid phases The most non-ideal case Case 1 Constant Rault's law Case with the least non-ideality Case 2 EOS Rault's law Medium non-ideal case
The properties appearing in the transport equations are determined by their averaging based on a phase volume-fraction [11].
@ @t(~v) +
@
@xj(~v~v) =
@p @xi + @ @xj @~v @xj +
@~v @xj
+ gi;
=Xn
q=1
qq; =
n
X
q=1
qq: (3)
Bulk Species Transport
Dynamic variations in the gas and liquid phase species concentrations are obtained by solving the unsteady state species mass balance equations written as Equa-tions 4 and 5, respectively:
@
@tgCig+ r:(g~vCig DiggrCig) = gNigl; (4)
@
@tlCil+ r:(l~vCil DilgrCil) = lNigl: (5)
Interphase Mass Transfer
The interphase mass transfer is related to the diusion at the interface that is related to the concentration gradients at the interface, too [12]. The concentration gradient of species in each phase was approximated using a Finite Dierence approach. In fact, the mass transfer coecient, based on the Film theory, is originally obtained through this approach. According to this approach, various elements of concentration gradients of phase `q' can be obtained as follows:
@Ciq
@xj
Ciq Ciq
xj ;
where Ciq is the concentration of the ith component in
phase q right at the interface and C
iqis the
concentra-tion of this component when phase q is at equilibrium with the other phase in the mixture (Figure 1). This is based on the fact that in a multiphase system, they are assumed to be at equilibrium right at their interface. Some authors assumed that vapor and liquid at the interface do not reach saturation and, hence, an eciency term (between 0.2 and 0.7) has been introduced. As good experimental results are lacking for the present study, the eciency term has been assumed to be 1 [9].
For a mixture containing vapor and liquid, the equilibrium concentration of various components can be obtained through isothermal ash calcula-tions [13,14]. Details of ash calculation algorithms and equations are given in Appendix.
Figure 1. Schematic of two-phase cell and equilibrium at the interface.
The concentrations of various species in vapor and liquid phases are obtained based on Equations 4 and 5, respectively. Having obtained equilibrium concentrations, one can calculate the ux of species transfer (Niq) and the rate of inter-phase mass transfer (Spq, which is the source term of Equation 1), through
Equations 6 to 8, respectively, in which Mi is the
molecular weight for the ith species. The calculated ux for component `i' (Nq
i) in one phase is a source
or sink for the same component in the other phase, because there is no accumulation at the interface.
Niq= DimCiq C iq
zj ; (6)
Nip= Niq; (7)
Spq= n
X
i=0
NiqMi: (8)
Simulation Procedure
The transport equations (Equations 1 and 3-5) were discretized by a control volume formulation [15]. The UPWIND method was used for discretization. A segre-gated implicit solver method with implicit linearization was used to solve discretized momentum equations. These equations have been obtained through appli-cation of the rst-order upwind method on Equa-tion 3, and for pressure velocity coupling, the SIMPLE method has been used [15]. For the pressure equation, the pressure staggering option (PRESTO) method was used [15].
The structure of the program code is outlined in Figure 2 and explained below. The program rst reads the structured data from the pre-processing section (in which the mesh representing the equipment has been built) before it goes into two nested iteration loops. Inner loop iterations are performed within each
Figure 2. Diagram illustration of program code structure.
time step using the equations corresponding to the discretized version of the proposed model while the outer loop goes through simulation times until it gets to the nal time or steady state; whichever happens sooner. At each time step, before going into the inner loop, the uid properties in each cell are calculated.
In the inner loop, all the discretized equations are solved in three steps. In the rst step, the physical properties, such as density, are updated based on the current solution. If the calculation has just begun, the uid properties will be updated based on the initialized solution. In the second step, the ash calculation is performed in order to obtain equilibrium concentrations based on which the source terms of the species concentrations and continuity equations are obtained. In the third step, equations of continuity and momentum are solved and after obtaining the velocity and pressure elds, equations corresponding to species concentration are solved in order to obtain the proles of the concentration of various species. In this step, with the help of the Eulerian-Eulerian approach (VOF approach), the trajectory of interface between two phases (liquid and gas) is determined. At the end of this step, convergence checking based on the norm of errors is done [14].
In order to get stable and meaningful results, the time step must be very small (in the order of 10 4 s).
In general, the time-stepping strategy depends on the
number of iterations by time step needed to ensure very low residual values (less than 10 7 for concentration
and 10 6 for momentum and continuity).
Computa-tional time is within 3-4 weeks for the two dimensional simulations. Calculations have been carried out on a 4GB RAM, 3.2 GHz CPU computer.
BENCHMARK FOR VALIDATION OF SIMULATION
We used experimental results that were taken for validation of the simulation in the Research Institute of Petroleum Industries (RIPI) [10]. A cylindrical vessel (Figure 3) lled with vapor and liquid hydrocarbon was selected as the benchmark. The liquid hydrocarbon was chosen to be pure Octane and the hydrocarbon in the gas phase was assumed to be Propane. Only the gas concentrations can be measured online due to the impossibility of the liquid phase measurement. The system was connected to a Gas Chromatography (GC) via two lines. Gas sampling is done automatically every 20 minutes. The location of sampling points was 20 cm from the top of the vessel.
RESULTS AND DISCUSSION
Figures 4 and 5 show the initial conditions of the sim-ulations for Cases 1 and 2 at which the concentration of octane in the gas phase and propane in the liquid phase is set to zero. It was also assumed that there is no movement in the system and, hence, the velocity was set to zero for the whole domain.
As time goes on, species are transferred between phases. This leads to a time varying concentration prole in both phases and a general velocity eld for
Figure 4. Contour of octane concentration (mol/liter) at t = 0:0 second (Cases 1 and 2).
Figure 5. Contour of propane concentration (mol/liter) at t = 0:0 second (Cases 1 and 2).
the whole uid both of which originated from the interphase mass transfer. The simulation results for concentration proles and velocity elds at certain times, for both cases, are shown in Figures 6 to 11. As seen in these gures, specic colors do not refer to the same levels in Cases 1 and 2 in a particular gure, because the variations of variables are dierent in the two cases and the colors are generated automatically by software. As Figures 7 and 10 show, octane was transferred from the liquid phase to the gas phase, and
Figure 6. Contours of velocity (m/sec) at t = 185 seconds (Cases 1 and 2).
Figure 7. Contour of octane concentration (mol/liter) at t = 185 seconds (Cases 1 and 2).
Figure 8. Contour of propane concentration (mol/liter) at t = 185 seconds (Cases 1 and 2).
Figure 9. Contour of velocity (m/sec) at t = 3500 seconds (Cases 1 and 2).
the concentration of octane in the liquid was decreased, whereas the concentration of octane in the gas was increased.
On the other hand, Propane dissolved in the liquid phase, which led to its concentration decrease in the gas phase. It can be seen, right at the interface, that the Propane concentration has its least value for the gas phase and the largest value for the liquid phase (Figures 8 and 11).
As seen in Figures 6 and 9, since the dissolution of octane in gas has not been considered in the calculation of the gas phase density (Case 1), velocity elds were dierent for Cases 1 and 2. Thus, bulk species con-centrations in gas and, consequently, gradients in the gas-liquid interface were solved dierently. Therefore,
Figure 10. Contour of octane concentration (mol/liter) at t = 3500 seconds (Cases 1 and 2).
mass uxes of species were dierent too. Also, in both cases, phase equilibrium calculations were done using Raoult's law, in spite of Case 0. Therefore, the errors are intensied, in comparison with Case 0. The inter-phase mass transfer for Cases 0, 1 and 2 (at t = 3500 s) were 0.07, 0.05 and 0.009 kg/m3.s, respectively.
At t = 3500 s, the proles of species concentra-tions along with the vessel length for Cases 0, 1 and 2 were presented (Figures 12 and 13). As a result of ideal assumptions, the species mole fraction in Cases 1 and 2 go away from the mole species fraction in Case 0. Due to the dierence in the rate of species transfer between the phases for these cases, the distribution of species concentration was dierent.
In order to see the deviation of the mentioned cases from experimental data, quantitative compar-isons were done between data obtained for Octane con-centration in the gas phase [10] and the corresponding simulated results shown in Figure 13. Since it was
Figure 11. Contour of propane concentration (mol/liter) at t = 3500 seconds (Cases 1 and 2).
Figure 12. Prole of octane concentration in gas phase (mol/liter) at t = 3500 seconds (Cases 1 and 2).
Figure 13. Prole of propane mole fraction in liquid phase at t = 3500 s (Cases 0, 1 and 2).
not possible to use the GC for the dynamic measure-ment of more than one point, only ve experimeasure-mental data have been obtained and compared against their corresponding points obtained by simulation. Only gas concentrations can be measured online due to the impossibility of the measurement liquid phase. As illustrated in Figure 14, the maximum amount of dierence between simulations and experimental data would occur at the start of the simulation (t = 0). The mentioned dierence was due to the delay in the Gas Chromatograph injection during the xing of the system pressure. Because of using Raults law in equilibrium calculation, instead of the fugacity coe-cient denition for the liquid and gas phases, in both cases, the ux of species is calculated incorrectly. As seen in Figure 13, the dierences in the cases between simulations and experiments intensied, in comparison with previous studies [10]. In Case 1, since the densities of phases were assumed constant, the eects of dissolved components on the density of phases and the buoyancy eect are not considered. Consequently, incorrect velocity elds and species concentrations, in addition to the mass transfer ux, are predicted.
Table 2 shows the simulated and measured con-centration of Octane in the gas phase, along with their relative dierences. As this table shows, the errors in the Octane mole fraction in the gas phase at all times are less than ve percent, while all non-idealities were considered (Case 0). Since the system is not at equilibrium and the mass transfer is simulated based on the CFD approach, and no empirical correlation has been used in the simulation, these small errors can be used as a rationale for the accuracy of the simulation results including the velocity and gas phase volume fraction proles. The errors in Cases 2 and 1 were higher: (21-31%) and (38-65%), respectively. These errors show that such simplications in similar modeling cases lead to wrong predictions.
Figure 14. Comparison of experimental and simulations results.
Table 2. Comparison between simulation and experimental data and relative errors.
Time Experimental Case 0 Case 2 Case 1 Error (Case 0) Error (Case 2) Error (Case 1)
0 0.018 0 0 0 100 100 100
14 0.0528 0.0515 0.036 0.018 2.46212121 31.818182 65.909091 26 0.1273 0.128234 0.1 0.05 0.73369992 21.445405 60.722702 41 0.1983 0.2047 0.147 0.09 3.22743318 25.869894 54.614221 56 0.26053 0.2676 0.238 0.16 2.713699 8.6477565 38.586727
CONCLUSIONS
The aim of this paper was to study the importance and eects of considering non-ideal thermodynamics in the simulation, which were presented in previous studies [10].
For this purpose, the benchmark was simulated by using a numerical method based on a macroscopic model and the nite volume method. In this simu-lation, non-ideality was not considered in the cases. Raoult's law was used in the equilibrium calculation instead of the fugacity coecient denition for the liquid and gas phases, in both cases, and in Case 1 the density of phases was assumed constant.
Quantitative validation of the simulated system with experimental data was based on the online anal-ysis of gas phase ow by Gas Chromatography. The predictions in both cases were compared with the experimental measurements and the simulation data in our previous work. It was found that the dierence between the gas species concentrations in experiment and simulation increased by the assumption of more ideality. The results also indicated that the assumption of constant density in simulation had a major role in diverting from experimental data (Figure 14). It is worth mentioning here that the closure for the mass transfer is not as mature as the closures used for the hydrodynamics. However, we became condent that if a more accurate closure for the mass transfer by considering non-ideality were to be applied, the present model would give a closer comparison with the experimental investigation, as has been shown in this study.
The model in our previous work was based on the Eulerian-Eulerian approach. Also, it combined hy-drodynamics, mass transfer and mixture non-ideality. The model and results presented in this work would be useful for extending the application of CFD based models for simulating large multiphase reactors. NOMENCLATURE
Cig gas species concentration
Cil liquid species concentration
C
i Equilibrium species concentration
CgT total concentration of gas
Dim diusion coecient of species i in
mixture
Ki equilibrium constant t
Mi molecular weight
Ni ux of species transfer
P pressure
V F Equilibrium volume fraction
~v velocity vector
Spq rate of interphase transfer
q phase density
mixture density
q phase viscosity
mixture viscosity
q volume fraction of each phase
Pdew dew point pressure
Pbub bubble point pressure
Zi mole fraction of species i in mixture
yi gas phase mole fraction
xi liquid phase mole fraction
REFERENCES
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5. Bezzo, F., Macchietto, S. and Pantelides, C.C. \A general methodology for hybrid multizonal/CFD mod-els. Part I: Theoretical framework", Computers and Chemical Engineering, 28, pp. 501-511 (2004). 6. Zauner, R. and Jones, A.G. \Scale-up of continuous
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7. Krishna, R. and Van Baten, J.M. \CFD modeling of bubble column reactor carrying out consecutive reaction (A ! B ! C)", Chem. Eng. Thechnol., 27, pp. 67-75 (2004).
8. Breach, M.R. and Ansari, N. \Modeling non-ideal vapor-liquid phase equilibrium, mass and energy trans-fer in a binary system via augmentation of compu-tational uid dynamical methods", Comput. Chem. Eng., 31, pp. 1047-1054 (2007).
9. Banerjee, R. \Turbulent conjugate heat and mass transfer from the surface of a binary mixture of ethanol/iso-octane in a countercurrent stratied two-phase ow system", International Journal of Heat and Mass Transfer, 51, pp. 5958-5974 (2008).
10. Irani, M., Bozorgmehry, R.B., Pishvaie, M.R. and Tavasoli, A. \Investigating the eects of mass transfer and mixture non-ideality on multiphase ow hydrody-namics using CFD methods", IJCCE (2009).
11. Hirt, C.W. and Nichols, B.D. \Volume of Fluid (VOF) method for the dynamics of free boundaries", J. Comp. Phys., 201, p. 39 (1981).
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15. Patankar, S.V., Numerical Heat Transfer and Fluid Flow, Taylor and Francis (1980).
16. Poling, B.E., Prausnitz, J.M. and O'Connell, J.P., The Properties of Gases & Liquids, 5th Ed., McGraw-Hill, New York (2000).
APPENDIX
Algorithm for Calculation of C
A ow chart has been sketched to describe the algo-rithm graphically (Figure A1). However, the routine for calculation of C is as follows:
1. In each two-phase cell, calculate dew-point (Pdew)
and bubble-point (Pbub) pressures.
2. If pressure of system is greater than Pbubthen:
Xi= Zi:
Figure A1. Flow chart of equilibrium calculation.
3. If pressure of system is less than Pdew then:
Xi= Zi=Ki:
4. If pressure of system is greater than Pdew and
less than Pbub then perform ash calculation (the
Antoine vapor pressure correlation was was used in equilibrium calculation):
ln Psat= a + b
T + c+ d ln(T ) + eTf; Ki(Ti) = P
sat i (T )
P ;
xi=1 + V F (Kzi i 1);
yi= 1 + V F (KziKi i 1);
F (V ) =Xyi
X xi= 0;
Cg Total=
n
X
j=1
Cg j;
C
Binary Diusivity
Fuller's method was used to determine diusivity:
DAB= 0:0143 T
1:75
P M0:5 AB
h (Pv)13
A+ (
P v)13
B
i2;
2 MAB =
1 MA +
1 MB:
P
v is found for each component by summing atomic diusion volumes given in [16].
BIOGRAPHIES
Mohammad Irani is a researcher at the RIPI in Tehran, Iran. His main research activities include CFD, simulation of multiphase/multicomponent and xed-bed reactors. He received both his MS and PhD degrees in Chemical Engineering from Sharif University of Technology, Tehran, Iran.
Ramin Bozorgmehry Boozarjomehry received his MS and PhD degrees in Chemical Engineering from Sharif University of Technology, Tehran, Iran, and The University of Calgary, Canada, in 1990 and 1997, respectively. Since then, he has been with the Chemical & Petroleum Engineering Department of Sharif Uni-versity of Technology. His research interests include: Process Simulation, Control and Optimization. Mahmoud Reza Pishvaie received both MS and PhD degrees in Chemical Engineering from Sharif University of Technology, Tehran, Iran, in 1991 and 2001, respectively. Since then, he has been at the Chemical Engineering Department of Sharif University of Technology, where he is currently Associate Pro-fessor. Since 2007, he has been Vice Chairman of the Research Aairs of the department and a member of the Research Council at SPGC, Asalooyeh. His research interests include: Optimization, Control and Simulation of Processing Plants.
